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What is the true significance of the constant "c" that we add after we integrate a curve without applying limits?

Srivatsav
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  • It means there are various functions which differentiated would give the same result of your original integrand. These various functions differ by constant amounts (which would disappear in the differentiation). – Henry Oct 29 '21 at 13:36
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    Have you read this: Purpose Of Adding A Constant After Integrating A Function? –  Oct 29 '21 at 13:39
  • For example $\frac{d}{dx} \big(\sin^2(x)\big) = \frac{d}{dx} \big(-\cos^2(x) \big)= \frac{d}{dx} \big(-\frac12\cos(2x) \big)= 2 \sin(x) \cos(x)$ so you cannot say precisely which is the integral of $2 \sin(x) \cos(x)$. Saying $\int 2 \sin(x) \cos(x) , dx = \sin^2(x) +c$ covers all these cases with $c$ being $0$ or $-1$ or $-\frac12$ to cover those three examples – Henry Oct 29 '21 at 13:43
  • The significance is that parallel lines exist. – PM 2Ring Oct 29 '21 at 13:49

2 Answers2

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Actually, talking in $\Bbb R^2$ after integrating when we obtain a function $g(x)+c$ the set I={$g(x)+c | \forall c\in \Bbb R $} it indicates the entire Family of Curves . But when we fix the quantity 'c' to a fixed real number , we are actually indicating a specific curve out of the Family of Curves .That's the role of the integrating constant.

MAHI
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Integration is reverse process of Differentiation. Suppose, f(X) = 2X+ 3

f'(X) = 2 ---->(1)

Again, f(X) = 2X , f'(X) = 2---->(2)

Now, if you integrate 2 , we take integral value as

2x+c (because of What I mentioned above. Since, derivative of constant term is 0)

Hence, we don't know wheather it has constant term or not ! That's why we take a arbitrary constant C.

Nope
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